We should already be familiar with the general principle for series circuits statingthat individual voltage drops add up to the total applied voltage, but measuringvoltage drops in this manner and paying attention to the polarity (mathematicalsign) of the readings reveals another facet of this principle: that the voltagesmeasured as such all add up to zero:

T

his principle is known as

Kirchhoff's Voltage Law

(discovered in 1847 by Gustav R.

K

irchhoff, a German physicist), and it can be stated as such:

"

The algebraic sum of all voltages in a loop must equal zero

"

By

algebraic

, I mean accounting for signs (polarities) as well as magnitudes. By

loop

,I mean any path traced from one point in a circuit around to other points in thatcircuit, and finally back to the initial point. In the above example the loop wasformed by following points in this order: 1-2-3-4-1. It doesn't matter which point westart at or which direction we proceed in tracing the loop; the voltage sum will stillequal zero.

T

o demonstrate, we can tally up the voltages in loop 3-2-1-4-3 of thesame circuit:

T

his may make more sense if we re-draw our example series circuit so that allcomponents are represented in a straight line:It's still the same series circuit, just with the components arranged in a differentform.

N

otice the polarities of the resistor voltage drops with respect to the battery:the battery's voltage is negative on the left and positive on the right, whereas allthe resistor voltage drops are oriented the other way: positive on the left andnegative on the right.

T

his is because the resistors are

resisting

the flow of electronsbeing pushed by the battery. In other words, the "push" exerted by the resistors

against

the flow of electrons

must

be in a direction opposite the source of electromotive force.Here we see what a digital voltmeter would indicate across each component in thiscircuit, black lead on the left and red lead on the right, as laid out in horizontalfashion:If we were to take that same voltmeter and read voltage across combinations of components, starting with only R

1

on the left and progressing across the wholestring of components, we will see how the voltages add algebraically (to zero):

T

he fact that series voltages add up should be no mystery, but we notice that the

polarity

of these voltages makes a lot of difference in how the figures add. Whilereading voltage across R

1

, R

1

--R

2

, and R

1

--R

2

--R

3

(I'm using a "double-dash" symbol "--" to represent the

series

connection between resistors R

1

, R

2

, and R

3

), we see howthe voltages measure successively larger (albeit negative) magnitudes, because thepolarities of the individual voltage drops are in the same orientation (positive left,negative right).

T

he sum of the voltage drops across R

1

, R

2

, and R

3

equals 45 volts,which is the same as the battery's output, except that the battery's polarity isopposite that of the resistor voltage drops (negative left, positive right), so we endup with 0 volts measured across the whole string of components.

T

hat we should end up with exactly 0 volts across the whole string should be nomystery, either. Looking at the circuit, we can see that the far left of the string (leftside of R

1

: point number 2) is directly connected to the far right of the string (rightside of battery: point number 2), as necessary to complete the circuit. Since thesetwo points are directly connected, they are

electrically common

to each other. And,as such, the voltage between those two electrically common points

must

be zero.

K

irchhoff's Voltage Law (sometimes denoted as

KVL

for short) will work for

any

circuit configuration at all, not just simple series.

N

ote how it works for this parallelcircuit:Being a parallel circuit, the voltage across every resistor is the same as the supplyvoltage: 6 volts.

T

allying up voltages around loop 2-3-4-5-6-7-2, we get:

N

ote how I label the final (sum) voltage as E

2-2

. Since we began our loop-steppingsequence at point 2 and ended at point 2, the algebraic sum of those voltages willbe the same as the voltage measured between the same point (E

2-2

), which of course must be zero.

T

he fact that this circuit is parallel instead of series has nothing to do with thevalidity of

K

irchhoff's Voltage Law. For that matter, the circuit could be a "blackbox" -- its component configuration completely hidden from our view, with only aset of exposed terminals for us to measure voltage between -- and